The Schinzel–Zassenhaus conjecture on houses of algebraic integers

Let $\alpha$ be an algebraic integer of degree $d$, with conjugates $\alpha=\alpha_1,\ldots,\alpha_d$. Its house is $\house{\alpha}=\max_{1\leq i\leq d}|\alpha_i|$. An algebraic integer is a root of unity exactly when its house is $1$. Schinzel–Zassenhaus conjecture. There is a constant $c>0$ such that, if $\alpha$ is not a root of unity, then

$$\house{\alpha}\geq 1+\frac{c}{d}.$$

This conjecture asks for a uniform lower bound on how close the house of a noncyclotomic algebraic integer can be to $1$. The source attributes it to Schinzel and Zassenhaus; no resolution is given here.